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        <title>Compactness Theorem</title>

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                <h1 id="title" titleSize="">
                    Compactness Theorem
                </h1>
            
            <h1 id="statement-and-proofs">Statement and Proofs</h1>
<div class="env" id=""><img class="icon noSelect listenDark" src="https://zhaoshenzhai.github.io/mathwiki/css/fa/theorem.svg"><b class="envTitle">Theorem (Compactness Thoerem; Gödel 1930 &amp; Maltsev 1936). </b>Let $L$ be a first-order language. If an $L$-theory $T$ is finitely-satisfiable (in the sense that every finite subtheory $\Delta\subseteq T$ is satisfiable), then $T$ is satisfiable.</div>

<p><div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof (from Completeness).</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent"><p>Every $L$-theory is <em>syntactically-compact</em>, in the sense that $T$ is consistent iff every $T$ is finitely-consistent.</p>
<blockquote>
<p>Indeed, if $\phi$ witnesses inconsistency of $T$, then there are finite subtheories $\Delta_0,\Delta_1\subseteq T$ such that $\Delta_0\proves\phi$ and $\Delta_1\proves\lnot\phi$. Their union $\Delta\coloneqq\Delta_0\cup\Delta_1$ is then an inconsistent finite subtheory of $T$, a contradiction.</p>
</blockquote>
<p>The <a href=https://zhaoshenzhai.github.io/mathwiki/completeness_theorem.md class="internalLink proved_by" title="Completeness Theorem" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/completeness_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Completeness Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/completeness_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Completeness Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/completeness_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/completeness_theorem.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:38:07-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Completeness Theorem&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/completeness_theorem&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Completeness Theorem</a> then applies, since if $T$ is finitely-satisfiable, it is finitely-consistent, and hence consistent by the above.<span style="float:right;">$\blacksquare$</span></p>
</span></div>
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<p><div class="collapsibleContainer" id=""><i class="proofHeader collapsibleHeaderButton collapsibleHeader noSelect">Proof (with Ultraproducts).</i><span class="collapsibleHintText noSelect"><i> Click to expand...</i></span>

        <span class="collapsibleContent">Suppose w.l.o.g. that $T$ is infinite and let $\mc{D}$ be the collection of all finite-subtheories of $T$. For each $\Delta\in\mc{D}$, let $M_\Delta\models\Delta$, and let $X_\Delta\subseteq\mc{D}$ be the subcollection of all finite-subtheories of $T$ containing $\Delta$. Since $X_{\Delta_1}\cap X_{\Delta_2}=X_{\Delta_1\cup\Delta_2}$ for any two $\Delta_i\in\mc{D}$, the collection $F\coloneqq\l\{X\subseteq\mc{D}\st X\supseteq X_\Delta\textrm{ for some }\Delta\in\mc{D}\r\}$ is a filter, which extends to an ultrafilter $U\supseteq F$.
<br>
  We claim that the <a href=https://zhaoshenzhai.github.io/mathwiki/ultraproduct.md class="internalLink references" title="Ultraproduct" mathLink="" secID="" secDisplay="" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/ultraproduct.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ultraproduct&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ultraproduct&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ultraproduct&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ultraproduct&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/ultraproduct.md&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ultraproduct&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ultraproduct&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">ultraproduct</a> $M\coloneqq\prod_{\Delta\in\mc{D}}M_\Delta/U$ models $T$. Indeed, for any $\phi\in T$, we have $M_\Delta\models\phi$ for all $\Delta\in X_{\l\{\phi\r\}}$, and thus $X_{\l\{\phi\r\}}\subseteq\l\{\Delta\in\mc{D}\st M_\Delta\models\phi\r\}$. Since $X_{\l\{\phi\r\}}\in U$, we have $\l\{\Delta\in\mc{D}\st M_\Delta\models\phi\r\}\in U$, and hence $M\models\phi$ by <a href=https://zhaoshenzhai.github.io/mathwiki/ultraproduct.md/#los_theorem class="internalLink proved_by" title="Ultraproduct" mathLink="" secID="los_theorem" secDisplay="Łoś’s Theorem" onmouseover="previewSide(&#34;https://zhaoshenzhai.github.io/mathwiki/ultraproduct.md/#los_theorem&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ultraproduct&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ultraproduct&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onmouseleave="clearPreviewSide({&#34;Date&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ultraproduct&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ultraproduct&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});" onclick="updateCurrentSide(event, &#34;https://zhaoshenzhai.github.io/mathwiki/ultraproduct.md/#los_theorem&#34;, {&#34;Date&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;Lastmod&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;PublishDate&#34;:&#34;2024-12-15T16:27:53-05:00&#34;,&#34;ExpiryDate&#34;:&#34;0001-01-01T00:00:00Z&#34;,&#34;Aliases&#34;:null,&#34;BundleType&#34;:&#34;&#34;,&#34;Description&#34;:&#34;&#34;,&#34;Draft&#34;:false,&#34;IsHome&#34;:false,&#34;Keywords&#34;:null,&#34;Kind&#34;:&#34;page&#34;,&#34;Layout&#34;:&#34;&#34;,&#34;LinkTitle&#34;:&#34;Ultraproduct&#34;,&#34;IsNode&#34;:false,&#34;IsPage&#34;:true,&#34;Path&#34;:&#34;/ultraproduct&#34;,&#34;Slug&#34;:&#34;&#34;,&#34;Lang&#34;:&#34;en&#34;,&#34;IsSection&#34;:false,&#34;Section&#34;:&#34;&#34;,&#34;Sitemap&#34;:{&#34;ChangeFreq&#34;:&#34;&#34;,&#34;Priority&#34;:-1,&#34;Filename&#34;:&#34;sitemap.xml&#34;,&#34;Disable&#34;:false},&#34;Type&#34;:&#34;page&#34;,&#34;Weight&#34;:0});">Łoś’s Theorem</a>.<span style="float:right;">$\blacksquare$</span></span></div>
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<p><strong>Remark.</strong> Thus, to show that a theory $T$ is satisfiable, it suffices to show that an arbitrary finite subtheory $\Delta\subseteq T$ is satisfiable. To further simplify this, we can take conjunctions and show that the single sentence $\phi\coloneqq\bigwedge\Delta$ is satisfiable.</p>


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                December 15, 2024 | Zhaoshen Zhai

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